Is the twin prime conjecture independent of Peano Arithmetic?
Alessandro Berarducci, Antongiulio Fornasiero, Joel David Hamkins
TL;DR
This paper constructs an arithmetical statement equivalent to the twin prime conjecture that ZF proves is independent of PA, yet its truth remains unprovable in ZF or any trusted metatheory.
Contribution
It demonstrates the existence of a formal statement equivalent to the twin prime conjecture that is provably independent of PA within ZF, highlighting limitations in formal proof systems.
Findings
ZF proves F is independent of PA
F is equivalent to the twin prime conjecture
The truth of F cannot be established in ZF or any trusted metatheory
Abstract
We show that there is an arithmetical formula F such that ZF proves that F is independent of PA and yet, unlike other arithmetical independent statements, the truth value of F cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula F such that ZF proves that F is equivalent to the twin prime conjecture.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHistory and Theory of Mathematics · Logic, programming, and type systems · Computability, Logic, AI Algorithms